Plane

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Introduction:

The plane is one of the most common surfaces encountered. This is because it is easy to parameterized and can be parameterized many different ways. Also, many theorems in math rely on a surface having a tangent plane defined at every point of its surface.

Definition:

Properties:

Partial Derivatives:

Because z is a function of x and y, we can take partial derivatives:

     

     

Tangent Planes:

At u = u0, v = v0, the tangent plane to the surface is parameterized by:

Infinitesimal Area:

The infinitesimal area of a patch on the surface is given by

Gaussian Curvature:

Mean Curvature: